Can we change the operating name of our business while keeping the legal name the same? How?

Yes. That's called a "fictitious business name" or an "alias" or "doing business as (DBA)" depending on what state you are in. Should be a fairly easy filing option available from your Secretary of State. If you can't figure it out on your own any attorney could do that for you, although you should be able to do on your own.

If I remember correctly, this goes through your filing lawyer. When you incorporated you needed a law firm to sign off your documents, this name change would go through them. Business as names can vary from the corporate tax name but you have to advertise them to satisfy the fcc.

Are you starting a new business unit? If so then just create an llc and that'll take care of it. Or vice versa.

The laws of business filings with respect to incorporation, names, etc. will vary from state to state. The best definitive resource (other than a lawyer) is your Secretary of State web site for your state of incorporation. In my experience, I have filed a registered name which created a DBA (doing business as) name. I was able to advertise and operate on this name with the need use DBA for certain legal documents, such as checks. On invoices, it would put my DBA name and underneath put (a Division of ParentCo, Inc.).

With any name change, there might be ramifications involving domain names. Many you might not think of.

If you'd like to discuss choosing a name or evaluate relevant domains, then let's talk.